Is this a theorem you have made up? If so, please prove it, or give a source for the theorem and its proof.

The only theorem I have seen is f is uniformly continuous on [a,b] if f is continuous on [a,b].

We are all aware of the epsilon/delata requirement.

The OP can be proved for a finite closed interval.

None of the proofs so far are correct. Do you have a proof?